76.6.10 problem 10

Internal problem ID [17394]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 3. Systems of two first order equations. Section 3.2 (Two first order linear differential equations). Problems at page 142
Problem number : 10
Date solved : Monday, March 31, 2025 at 04:12:44 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=-x \left (t \right )+2 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=-2 x \left (t \right )-y \left (t \right ) \end{align*}

✓ Maple. Time used: 0.144 (sec). Leaf size: 44

ode:=[diff(x(t),t) = -x(t)+2*y(t), diff(y(t),t) = -2*x(t)-y(t)]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{-t} \left (c_2 \cos \left (2 t \right )+c_1 \sin \left (2 t \right )\right ) \\ y \left (t \right ) &= {\mathrm e}^{-t} \left (\cos \left (2 t \right ) c_1 -\sin \left (2 t \right ) c_2 \right ) \\ \end{align*}

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 51

ode={D[x[t],t]==-x[t]+2*y[t],D[y[t],t]==-2*x[t]-y[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to e^{-t} (c_1 \cos (2 t)+c_2 \sin (2 t)) \\ y(t)\to e^{-t} (c_2 \cos (2 t)-c_1 \sin (2 t)) \\ \end{align*}

✓ Sympy. Time used: 0.094 (sec). Leaf size: 44

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(x(t) - 2*y(t) + Derivative(x(t), t),0),Eq(2*x(t) + y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = C_{1} e^{- t} \sin {\left (2 t \right )} + C_{2} e^{- t} \cos {\left (2 t \right )}, \ y{\left (t \right )} = C_{1} e^{- t} \cos {\left (2 t \right )} - C_{2} e^{- t} \sin {\left (2 t \right )}\right ] \]